H3 Relativistic Kinematics

Light Clock: A beam of light reflected between two parallel mirrors that is used to measure time is called a light clock.

From: http://www.patana.ac.th/secondary/science/anrophysics/relativity_option/commentary.html

As can be seen, to Jill, her light clock appears to have spanned exactly 10 seconds no matter what her speed. This is because the light clock is stationary with respect to her and the beam of light only has vertical motion not horizontal motion. But if Jack observes Jill's clock, the beam of light from the source is not only moving upwards but also to the right. This means that it has a greater distance to travel than the beam of light approaching the mirror from his light source. As the speed of light is constant, this must mean Jill's clock must be moving slower according to Jack to compensate for the increased distance that the light beam has to travel.

Proper Time is the time as measured in the frame of reference where the event it occurring. This happens to be the shortest possible time that can be recorded for an event to occur.
Using the above example, both Jack and Jill will measure proper time using their clocks as they are in the frame of reference in which the event is occurring.

Time Dilation Derivation:

Yeah...you can't really see what's written.
This is what is written:

HL Physics Hienemann, Chris Hamper

Gamma (in the video) is called the Lorentz factor.
The speed of light must be less than c, as otherwise the denominator would be 0 and time would be infinitely long. Therefore, we always measure the time taken for an event to occur in different frame of reference that has relative motion to be longer than the time taken for that event as measured by someone in that frame of reference. Time is said to dilate.

The variation of the Lorentz factor as the velocity of the object varies is shown below:

Graph Data

Time Dilation problems:

Proper Length:Proper length of an object is the length of an object as measured by an observer in the same frame of reference as the object. This is the longest possible distance of the object that can be measured. As on object approaches the speed of light, its length appears to decrease in the direction of its motion.

If an astronaut is moving between stars close to the speed of light, then the distance between the stars appears to contract which means that the astronaut feels that the distance be travels is shorter and he covers it in a shorter time than what it appears to be from earth.